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Re: Coupled Diff Eqs or Poisson Eq, is symbolic solution

• To: mathgroup at smc.vnet.net
• Subject: [mg102858] Re: [mg102844] Coupled Diff Eqs or Poisson Eq, is symbolic solution
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Mon, 31 Aug 2009 06:33:34 -0400 (EDT)
• Organization: Mathematics & Statistics, Univ. of Mass./Amherst
• References: <200908301006.GAA20836@smc.vnet.net>
• Reply-to: murray at math.umass.edu

Your post is essentially unreadable because of the embedded (hex?) codes 
such as "=E2", "=CF", etc.  Please use plain ASCII and remove all such 
codes.

Neelsonn wrote:
> Guys,
> 
> (This is the 3rd time I am trying to post this; I apologize for any
> duplicate)
> 
> I've have just installed Mathematica and have some tasks to accomplish
> using it. I've spent some time trying to find similar problems at
> Wolfram's website, but not success so far. So I am posting here for
> the first time (unless someone, please, point me a similar post or
> documentation)
> 
> Here's what I need to solve:
> (Poisson)
> 
> =E2=88=87^2(x,y) = Rs * J(x,y)
> 
> 
> for two cases:
> 
> i)
> 
> =CF=88 = 0 for x = 0, x = a, y = b;
> 
> =E2=88=82=CF=88/=E2=88=82y = 0 for y = 0.
> 
> and
> 
> ii)
> 
> =CF=88 = 0 for x = 0, x = a, y = 0, y = b.
> 
> 
> Some side notes:
> 
> - Physically speaking, for both cases I would like to know how the
> electrostatic potential (=CF=88) will be distributed on a rectangular shape
> (a,b) when it has grounded electrodes on the three edges (case i) and
> grounded electrodes surrounding all four edges (case ii).
> 
> - The rectangular shape resembles a resistive material, that comes the
> Rs (sheet resistance) and J(x,y) is the current that is going to be
> distributed on the surface of this geometry as well. In my case J(x,y)
> = exp(V(x,y)). An "arrow plot" showing the current distribution will
> be also interesting.
> 
> - Eventually, once the solution =CF=88(x,y) is found, the electric field E
> (x,y) = - =E2=88=87=CF=88 and the total current flow J = 1/=CF=81 * E, =
> where =CF=81 is the
> resistivity (ohm.meter), is also interested
> 
> ----------
> 
> Now, my question is: Can Mathematica handle such problem and
> boundaries like it is in order to solve it analytically (symbolic)? I
> haven't seen, in the examples, problems like this. I wonder if I will
> have to decouple
> 
> =E2=88=87^2(x,y) = Rs * J(x,y)
> 
> into first-order partial differential equations. Then,  a follow-up
> question that comes: can Mathematica do that automatically or I should
> pose the problem myself? For that, I've seen an example from the
> website that uses six first-order differential equations to solve the
> kinetics of some chemical reactions. But the problem was solved
> numerically and I would like to have an analytical equation as a
> result. So is it possible to find such analytical solution in case I
> have to use a system of first-order partial diff eqs?
> 
> (I am pretty sure that this isn't a difficult problem for those who
> Master Mathematica)
> 
> A final question or better yet, help needed: I would like to do all
> the above for a different shape, not a rectangule or square, but for a
> trapezoidal shape. I have no idea how to start and don't know how the
> boundaries will look like. I wonder if there is a way to draw such
> shape in Mathematica and graphically tells the software to solve it
> (like those "FEM softwares"...). That would be very easy! I would
> really appreciate any input here.
> 
> Thanks
> N
> 
> 
> 
> 

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

• References:
  • Coupled Diff Eqs or Poisson Eq, is symbolic solution possible?
    • From: Neelsonn <neelsonn@gmail.com>